Maximal Number of Edges in Geometric Graphs without Convex Polygons

Abstract

AbstractA geometric graph G is a graph whose vertex set is a set P n of n points on the plane in general position, and whose edges are straight line segments (which may cross) joining pairs of vertices of G. We say that G contains a convex r-gon if its vertex and edge sets contain, respectively, the vertices and edges of a convex polygon with r vertices. In this paper we study the following problem: Which is the largest number of edges that a geometric graph with n vertices may have in such a way that it does not contain a convex r-gon? We give sharp bounds for this problem. We also give some bounds for the following problem: Given a point set, how many edges can a geometric graph with vertex set P n have such that it does not contain a convex r-gon?A result of independent interest is also proved here, namely: Let P n be a set of n points in general position. Then there are always three concurrent lines such that each of the six wedges defined by the lines contains exactly \(\lfloor \frac{n}{6} \rfloor\) or \(\lceil \frac{n}{6} \rceil\) elements of P n .KeywordsGeneral PositionConvex PolygonSharp BoundStraight Line SegmentGeometric GraphThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.